Calculus Ch 5 1 Critical Points Max Mins

Calculus Ch 5. 1 Critical Points, Max Mins

CRITICAL POINT f’(c) = 0 or f’(c) is undefined c is in the INTERIOR of the domain !!

To find max & mins for f(x), 1. Find f ' (x). 2. Factor. Set = 0 to find critical points.

3. Use a NUMBER LINE for a VISUAL. Examine points to the left and right of CRITICAL POINTS. Mark + and – signs to show where graph rises & falls.

a b c d f ’(a) is vertical = undefine It’s a critical point.

a b c d f ’(b) is horizontal = 0 Critical point

a b c d f ’(c) is sharp = undefine Critical point

a b c d f ’(d) is not in the interior. No critical point.

First derivative: Y’ is + graph increases. Y’ is neg graph decreases. Y’ = 0 Possible max or min.

1 st deriv Find critical poin If f is negative to the left o and + to the right of c, you have a local min. f falls f rises + c

1 st deriv Find critical poin If f is + to the left of c and negative to the right of c, y have a local MIN. f rises f falls+ c

Determine max and mins Extreme Value Th: If f is continuous on [a, b], f has a global max and min on [a Also known as absolute

Extreme Value Th: global max at c on [a, b] if f(x) < f(c) for all x. global min at c on [a, b] if f(x) > f(c) for all x.

Graph Roots are x= -1 and x= 2 x intercepts -1 2

First Deriv test 0 2

neg + + 0 2

Possible max a min at 2. - graph + + graph falls rises 0 2

y= 2 (x+1)(x-2) roots -1, 2 nd 2 deriv tells about concavity -1 0

If c is a Critical number of f has a local minimum at if f(c) < f(x) for all x near c If c is a Critical number o f has a local maximum at if f(c) > f(x) for all x near c